Now, the arguments of [1]} produce skeleta with singularities satisfying the characterization of Definition REF . So without the uniqueness of Theorem REF , we would still be left to study the possible moduli of such singularities: it could happen that two skeleta built with the same combinatorics lead to different Weinstein manifolds. The uniqueness of Theorem REF  guarantees this is not the case: there is no moduli of the singularities arising, and indeed their geometry is uniquely specified by the combinatorics. (In fact, the situation is even better: thanks to the arboreal Darboux-Weinstein theorem proved in [1]}, any symplectic thickenings of the skeleton that induce equivalent orientation structures, a further combinatorial decoration on the skeleton, are themselves equivalent.) Thus pairing the results of the current paper with those of [1]} one is able to express polarized Weinstein manifolds in combinatorial terms.
